Properties

Label 405600.df
Number of curves $4$
Conductor $405600$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("df1")
 
E.isogeny_class()
 

Elliptic curves in class 405600.df

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
405600.df1 405600df4 \([0, -1, 0, -5308008, -4704550488]\) \(428320044872/73125\) \(2823683265000000000\) \([2]\) \(12386304\) \(2.5469\)  
405600.df2 405600df2 \([0, -1, 0, -2266008, 1269177012]\) \(33324076232/1285245\) \(49629057065640000000\) \([2]\) \(12386304\) \(2.5469\) \(\Gamma_0(N)\)-optimal*
405600.df3 405600df1 \([0, -1, 0, -364758, -57895488]\) \(1111934656/342225\) \(1651854710025000000\) \([2, 2]\) \(6193152\) \(2.2004\) \(\Gamma_0(N)\)-optimal*
405600.df4 405600df3 \([0, -1, 0, 1008367, -391564863]\) \(367061696/426465\) \(-131741766411840000000\) \([2]\) \(12386304\) \(2.5469\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 405600.df1.

Rank

sage: E.rank()
 

The elliptic curves in class 405600.df have rank \(1\).

Complex multiplication

The elliptic curves in class 405600.df do not have complex multiplication.

Modular form 405600.2.a.df

sage: E.q_eigenform(10)
 
\(q - q^{3} + 4 q^{7} + q^{9} - 4 q^{11} - 2 q^{17} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.

\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.